The Power Law As an Emergent Property

Memory & Cognition 2001, 29 (7), 1061-1068

The power law as an emergent property

RICHARD B. ANDERSON Bowling Green State University, Bowling Green, Ohio

Recent work has shown that the power function, a ubiquitous characteristicof learning, memory, and sensation, can emerge from the arithmetic averaging of exponential curves. In the present study, the forgetting process was simulated via computer to determine whether power curves can result from the averaging of other types of component curves. Each of several simulations contained 100 memory traces that were made to decay at different rates. The resulting component curves were then arithmetically averagedto produce an aggregatecurve for each simulation. The simulations varied with re- spect to the forms of the component curves: exponential, range-limited linear, range-limited logarithmic, or power. The goodness of the aggregatecurve’s fit to a power function relative to other functions increased as the amount of intercomponent slope variability increased, irrespective of component- curve type. Thus, the power law’s ubiquity may reflect the pervasiveness of slope variability across component functions. Moreover, power-curve emergence may constitute a methodological artifact,an explanatory construct, or both, depending on the locus of the effect.

The power function is generally accepted as an appro- by J. R. Anderson and Schooler (1991) and Wixted and priate mathematicaldescription of human performance in Ebbesen (1991). psychophysics (Stevens, 1971), in skill acquisition (J. R. Repeated confirmation of the power law has prompted Anderson, 1982; Logan, 1988; Neves & J. R. Anderson, theoreticalexplanationsfor the phenomenon.Buildingon 1981), and in retention (J. R. Anderson & Schooler, 1991; Newell and Rosenbloom’s(1981) analysis of skill acqui- R. B. Anderson, Tweney, Rivardo, & Duncan, 1997; Ru- sition and Estes’s (1956) discussion of curve averaging, bin, 1982; Wixted & Ebbesen, 1991). Power curves have R. B. Anderson and Tweney (1997) argued that the power been observed so frequently, and in such varied contexts, law might characterize forgetting at the aggregate level that the term “power law” is now commonplace (but see only, and that the forgetting of single memory traces, or Ru-bin & Wenzel, 1996, for some apparent violations of parts of memory traces, might proceed exponentially.The the power law). Figure 1 contains a representative curve essence of the argument is captured by the following hy- from R. B. Anderson et al. (1997), illustrating the decline pothetical example (more extensive analyses will be pre- in recall performance as a power function of length of the sented later). Imagine that the memory store containsthree retention interval. The function is represented mathemat- traces that decay according to the exponential equation ically as Y =(A)e BX, B <0, (2) Y =(A)X B, B <0, (1) where Y is retrieval probability, X is time, A is the value of where Y is performance level (e.g., recall), X is time (e.g., Y when X = 0, B is the slope (as in Equation1), and e is the retention interval), and A and B are parameters defining base of natural logs. Thus, for each trace, retrieval proba- the precise shape of a given power curve. Parameter B de- bility declines by a constant percentage, over each unit of termines the slope1 of the curve; larger values of B yield time. Supposefurther that each curve has a different slope, steeper slopes. Parameter A is an intercept that specifies and that the curves combine to form an aggregate curve, the value of Y when X equals 1.0. R. B. Anderson et al. fit equal to the arithmetic average of the three components. the data in Figure 1 to four common functions: linear, ex- Figure 2a, illustrates the example. Though composed of ponential,logarithmic, and power. In accordance with the exponentialcomponents,the aggregate curve itself is non- power law, the goodness-of-fit to the power function exponential;its percent of decline, or loss-percentage,de- (using R2 as the fit criterion) was greater than the fit to the creases over time. Other functions with this characteristic other three functions.Similar findingshave been reported include the power function and the logarithmic function. However, a regression analysis of the aggregate curve in Figure 2a indicatesthat the curve fits a power function (R2 = .99) better than it does a logarithmic function (R2 = .98), I thank William Batchelder, Robert Bjork, Bennet Murdock, In Jae an exponentialfunction (R2 = .87), or a linear function (R2 = Myung, David Rubin, Ryan Tweney, John Wixted, and Steven Sloman, .76). This kind of observation has led to the hypothesis for their helpful comments on the manuscript. Correspondence should be addressed to R. B. Anderson, Department of Psychology, Bowling that forgetting is fundamentally exponential, and that the Green State University, Bowling Green, OH 43403 (e-mail: randers@ empirical power function is an artifact of averaging across bgnet.bgsu.edu). component curves. Indeed, reanalysis of published data

bloom (1981) recognized the same problem with regard to the hypothesis that skill acquisition might obey a power law at the aggregate level, but obey an exponentiallaw at the component-skill level. The present paper offers a re- fined and expanded version of the exponential-component hypothesis. In this new account, component curves need not be exponential(as in R. B. Anderson & Tweney, 1997; Newell & Rosenbloom,1981; Wickens, 1998) to produce a power curve at the aggregate level. Instead, the power law may be an emergent property that arises naturally from intercomponentslope variability, irrespective of the components’ mathematical form.2 Consider the case of linear components. Of course, the Figure 1. Mean number of items recalled as a function of re- arithmetic average of purely linear componentswill itself tention interval (RI). Adapted from R. B. Anderson, Tweney, Ri- be linear. However, if one introduces the reasonable as- vardo, and Duncan (1997). The data are from the flat test- probability condition in Experiment 2 and are collapsed across sumption that forgetting is range limited (i.e., that retrieval five blocks of trials. Solid line shows best-fitting power function. probabilitynever falls below zero), then even linear components can producean aggregate power function.Figure 2b shows three hypotheticalrange limited components.Com- sets has shown that arithmetic averaging across trial posi- ponent 1 is a proper linear function (within the time range tion often exaggerates the fit of the aggregate curve to a of the example). However, Components 2 and 3 have suf- power function, relative to an exponentialfunction (R. B. ficiently steep slopes so that performance drops to zero and Anderson & Tweney, 1997). remains there as time approachesits maximum. The figure Though tenable, the exponential-component hypothe- also shows that the aggregate curve fits a power function sis raises the question of why forgetting should proceed better than it fits a linear, logarithmic,or exponentialfunc- exponentiallyat the trace level. Indeed, Newell and Rosen- tion (see Table 1 for the R2 values). Figures 2c and 2d

Figure 2. Aggregate forgetting curves derived by arithmetic averaging of three hypotheti- cal component curves. Each panel shows a different type of component curves: exponential, range-limited linear, range-limited logarithmic, and power. Solid lines show best-fitting power functions. POWER LAW 1063

Table 1 100 traces. To insure that all curves would be descending (as in Fig- Goodness-of-Fit (R2) of Aggregate Curves (in Figure 2) to Best- ure 2) rather than ascending, B was prevented from taking on a neg- Fitting Exponential (Exp), Linear (Lin), Logarithmic (Log), ative value. If B became negative as a result of adding a random de- and Power (Pow) Functions viate, then B was resampled. (Note that the above constraint on B is Mean R2 distinct from the range limitation of the linear and logarithmic func- Component Type Exp Lin Log Pow tions.) The 100 component curves were arithmetically averaged to produce an aggregate curve. In order to assess the effect of compo- Exponential .87 .76 .98 .99 nent slope variability on the form of the aggregate curve, the stan- Linear (range limited) .74 .65 .92 .95 dard deviation of the random deviate distribution (described above) Logarithmic (range limited) .74 .59 .90 .97 was manipulated across six levels: 0, 0.1, 0.2, 0.3, 0.4, and 0.5. The sim- Power .71 .60 .89 .96 ulation was replicated 500 times at each level of slope variability. Note—Functions were fit by entering appropriate log transforms into a Although the power law has been observed using various curve- linear regression algorithm. fitting methods, the most frequent method involves appropriately transformed data into a linear regression algorithm R. B. Anderson & Tweney, 1997). Thus, since the linear regression method defines show that an aggregate power curve may also result from (to a great extent) the power-law phenomenon, that method will be the averaging of logarithmic components (range limited, used in the present analyses. as with the linear case) or power components.3 Table 2 contains the mathematical representations of the four func- RESULTS AND DISCUSSION tion types. Note that range limitation is unnecessary for exponentialand power functionssince bothfunctionsnec- Although other fit indices are available, the present essarily maintain a positive value. simulations used R2 as the goodness-of-fit measure. R2 Taken together, the examples demonstrate that inter- was computed by linearizing the aggregate curves and component slope variability can yield aggregate power then submitting them to a simple linear regression. Lin- curves, irrespective of the components’forms (i.e., expo- earization was accomplishedby transforming the Y axisto nential, range-limited logarithmic, range-limited linear). log Y (to assess fit to an exponentialfunction)or by trans- Consequently, the ubiquity of the power law may reflect forming the X axis to log X (to assess logarithmic fit) by the widespread presence of slope variability. log transforming both axes (to assess power fit). Of To date, there has been no formal, mathematical proof course, no linearization (i.e., no transformation) was nec- of power-curve emergence.Laplaceseries expansions,for essary to assess goodness-of-fit to a linear function. example, show that the power function cannotbe formally The incidence of aggregate power curves was assessed expressed as the sum of exponentialcomponents(R. B. An- by counting the number of times, n (of 500), in which the derson & Tweney, 1997; Newell & Rosenbloom, 1981). aggregate curve’s fit to a particular function was superior Moreover, in the case of range-limited linear and range- to its fit to any of the three remaining functions. Table 3 limited logarithmic components, formal proof via differ- shows that the number of aggregate power curves in- ential calculus seems inapplicable, since not every com- creased as amount of intercomponentslope variability in- ponent is fully differentiable. Consequently, the present creased, thereby establishing that slope variability at the paper employs computationalsimulation, rather than for- component level can produce a power function at the ag- mal mathematical proof, to demonstrate the phenome- gregate level. In the case of power components, the ag- non’sgenerality. gregate curve was best fit by a power function, regardless of slope variability.Thus, although the presence of an ag- METHOD gregate power curve is consistent with the existence of power components, it is also quite consistent with the Each simulation contained 100 memory traces (components) that presence of other types of component curves. Table 4 decayed exponentially, linearly, logarithmically, or in accordance shows similar results for simulations in which the A pa- with a power function (in the linear and logarithmic cases, the func- rameter is set to .5 insteadof .9.Note, however,thatTable tions were range limited). All component curves were arbitrarily as- 4 shows power curves sometimes combining to form ag- signed a value of .9 for the A parameter (Table 2). The simulations did not include manipulation of the A parameter: It is already known gregate logarithmic curves. that variability in A cannot be a general cause of emergent power The aggregate power-curve phenomenon was also as- curves since, in the case of exponential components, variability in A sessed by calculating the mean goodness-of-fit (defined has no effect on the aggregate fit (R. B. Anderson & Tweney, 1997; Estes, 1956). Values for B (i.e., the slopes) were generated randomly via the fol- Table 2 lowing procedure. First, B was set to a level that would yield a per- Mathematical Equations Representing Exponential, formance level of .5 after 20 units of simulated elapsed time (B was Linear, Logarithmic, and Power Functions set to .029, .020, .133, and .195, for the exponential, linear, logarithmic, and power components, respectively). Next, in order to gen- Function Type Equation erate component curves with differing slopes, random deviates were Exponential Y =(A)e BX, B <0 added to the initial B. The deviates were sampled from a normal pop- Linear Y = A BX, Y >0,B >0 ulation with a mean of zero. The sampling procedure was repeated Logarithmic Y = A B logX , Y >0,B >0 B to produce 100 values of B, defining a forgetting curve for each of Power Y =(A)X , B <0 1064 ANDERSON

Table 3 Effect of Component Type and Intercomponent Slope Variability on the Number of Cases (n) in Which the Aggregate Curve Was Best Fit by an Exponential (Exp), Linear (Lin), Logarithmic (Log), or Power (Pow) Function Component Slope n Type Variability Exp Lin Log Pow Exponential .0 500 0 0 0 .1 58 0 442 0 .2 0 0 495 5 .3 0 0 68 432 .4 0 0 0 500 .5 0 0 0 500 Linear .0 0 500 0 0 (range limited) .1 0 0 464 36 .2 0 0 1 499 .3 0 0 0 500 .4 0 0 0 500 .5 0 0 0 500 Logarithmic .0 0 0 500 0 (range limited) .1 0 0 500 0 .2 0 0 0 500 .3 0 0 0 500 .4 0 0 0 500 .5 0 0 0 500 Power .0 0 0 0 500 .1 0 0 0 500 .2 0 0 0 500 .3 0 0 0 500 .4 0 0 0 500 .5 0 0 0 500 Note—The components’ A parameters are fixed at 0.9. here and throughoutas mean R2) of the aggregate curve to able (as noted earlier) but is also an important factor in the best-fitting exponential,linear, logarithmic,and power producing power curves from exponential components. functions. Figure 3 shows the effect of the amount of intercomponentslope variability (i.e., the standard deviation Other Kinds of Distributions of distribution of random deviates) on the mean R2 aver- Newell and Rosenbloom (1981) argued that it should aged across 500 replications. be possible to create power curves from uniformly (i.e., rec- Overall, the power fit became increasingly superior as tangularly distributed exponential-component slopes). intercomponent variability increased, irrespective of However, Wixted and Ebbesen (1997) suggested that component-curve type. Of course, in the case of power- Weibull-distributedexponentialsdo not readily yield aggre- curve components(Figure 3d), the aggregate curve was fit gate power curves. Table 5 provides simulationresults that perfectly by a power function when slope variability was confirm Wixted’spoint but that also demonstratethe ready zero. Still, the relative goodness of the power fit increased emergence of aggregate logarithmic curves and the emer- with slope variability.In sum, the simulation results for R2 gence of a singleaggregatepower curve. Moreover,Tables 5 were consistentwith the results for n (i.e., the frequency of and 6 demonstrate that range-limited linear and range-lim- aggregate power curves) and with the examples described ited logarithmiccomponents,whetherWeibulldistributedor in the introduction, all of which show that variable com- uniformly distributed, still yield aggregate power curves. ponent slopes can yield aggregate power curves.4 It should be noted that the present simulationsindicated Response Surface Analysis a level of power-fit superiority that exceedslevels reported The present paper focuses on traditionalcomputational by R. B. Anderson and Tweney (1997). In that study, aggre- simulation to show that arithmetically averaged compo- gate power curves emerged primarily when component– nent curves—whether exponential, range-limited loga- slope differences were large and systematic; random vari- rithmic, or range-limited linear—tend to yield power-like abilityhad only a slight impact on the aggregate power fit curves at the aggregate level. Very recently, Myung, Kim, (also note that R. B. Anderson & Tweney’swork dealt ex- and Pitt (2000) used another technique, response surface clusively with exponential component curves). The pres- analysis (RSA), to make methodologicalpoints concern- ent simulations are unique, however, in that B was con- ing power-curve emergence. (They studied emergence strained so that all componentcurves would be downward from exponential components—not from linear or loga- sloping. Apparently, such a constraint is not only reason- rithmic components.) POWER LAW 1065

Table 4 Effect of Component Type and Intercomponent Slope Variability on the Number of Cases (n) in Which the Aggregate Curve Was Best Fit by an Exponential (Exp), Linear (Lin), Logarithmic (Log), or Power (Pow) Function Component Slope n Type Variability Exp Lin Log Pow Exponential .0 500 0 0 0 .1 500 0 0 0 .2 500 0 0 0 .3 500 0 0 0 .4 471 0 0 29 .5 1 0 188 311 Linear .0 0 500 0 0 (range limited) .1 0 0 498 2 .2 1 0 173 326 .3 9 0 92 399 .4 –––– .5 –––– Logarithmic .0 0 0 500 0 (range limited) .1 0 0 497 3 .2 0 0 142 358 .3 0 0 12 488 .4 0 0 0 500 .5 0 0 0 500 Power .0 0 0 0 500 .1 0 0 0 500 .2 0 0 120 380 .3 0 0 52 448 .4 0 0 21 479 .5 0 0 6 494 Note—In the case of linear components, when the amount of slope variability was very large, there were cases in which the aggregate curve contained zeros, precluding the log transform method of curve fitting. The components’ A parameters are fixed at 0.5.

The RSA approach is applicable to the present work curves—the forgetting curves are only implicitly repre- and offers an alternative means of drawing conclusions sented in the figure, as pairs of Y1 Y2 values. Rather, the similar to those obtained via traditional simulation. surfaces represent the range of responses (Y values) that Following Myung et al. (2000), two-dimensional re- can be produced by a given function (i.e., a given type of sponse surfaces may be constructed by first simplifying trace-level forgetting curve). the equationsin Table 2 so that the A parameter is omitted The surfaces in Figure 4 can be used as analytictools to (or, equivalently,fixed at 1.0). For example, in the case of demonstrate power-curve emergence: For any of the four an exponentialfunction, surfaces one can visualize the arithmetic averaging of two componentcurves by choosingany two arbitrary points, i Y =(A)e BX , B <0 is simplified to Table 5 Y = e BX, B <0. Mean R2 and Number of Best Fits (n) to Exponential (Exp), Linear (Lin), Logarithmic (Log), and Power (Pow) The next step is to compute the value of Y (i.e., the “re- Functions, for Aggregate Curves With Weibull-Distributed B sponse” in “RSA”) for two arbitrary values of X (X1 =2 Components and X2 = 10), with B also set to an arbitrary value. The Fitted Function resulting values of Y are referred to as Y and Y . Thus Y 1 2 1 Component Exp Lin Log Pow and Y2 define a point in a two-dimensional space. Next, Type R2 nR2 nR2 nR2 n the operationis repeated for different valuesof –B—keep- Exp .95 0 .83 0 .99 499 .86 1 ing the same two values for X1 and X2 (in the present Lin .92 4 .71 0 .96 98 .98 398 analysis, B varies across 1,000 values, from .01 to 10.0). Log .82 0 .72 0 .96 0 .99 500 The result is a set of points that define a curve, or surface, Note—Mean R2 is for the entire set of 500 aggregate curves (for each in two-dimensional space (see Myung et al., 2000, for a component type). The Weibull population distribution has shape and discussion of the surface’s formal analytic properties). range parameters of 1.0 and 0.25, respectively, with a mean of 0.25 (sampled values were multiplied by 1.0) and a standard deviation of Figure 4 shows four such surfaces, constructed from 0.25. The uniform population has a mean and standard deviation of component-curvefunctionsthat excludeA as a parameter. 0.25 and 0.125, respectively. Linear and logarithmic component It is important to note that these surfaces are not forgetting curves are range limited. 1066 ANDERSON

Table 6 equal to zero; (3) when the range of B is extreme, the ag- Mean R2 and Number of Best Fits (n) to Exponential (Exp), gregate curve tends to fit a power function better than it Linear (Lin), Logarithmic (Log), and Power (Pow) fits other common functions, even if the power fit is not Functions, for Aggregate Curves With Uniformly 5 Distributed B Components very good (e.g., Point Q). Fitted Function Qualifiers Exp Lin Log Pow Component The present analyses assume that at the component 2 2 2 2 Type R nR nR nR n level, retention tends not to increase over time and tends Exp .95 0 .78 0 .98 498 .96 11 not to take on a negative value. Because of these restric- Lin .88 2 .57 0 .87 98 .99 498 Log .87 1 .75 0 .97 1 .99 499 tions, manipulation of interslope variability tends to alter Note—Mean R2 is for the entire set of 500 aggregate curves (for each the shape of the distributionof slopes. Thus, it is possible component type). The uniform population distribution has a mean and that slope-distribution shape is the proximal cause of standard deviation of 0.25 and 0.125, respectively. Linear and loga- power-curve emergence, with slope variability being an rithmic component curves are range limited. indirect or even noncritical factor. In principle, it may be possible to manipulate shape while holding variance constant, but then shape would still be confounded with other and j (each point in Figure 4 actually represents an entire measures of variability, such as range. The present work component curve) and interpolating a straight line be- addresses the shape issue to some degree by showing tween the two points. The point at the center of the line power-curve emergence in Weibull and rectangulardistri- represents the arithmetic average of the two curves repre- butions(in additionto range-limitednormal distributions). sented by arbitrary points i and j. Thus, Figure 4 can be Futhermore, R. B. Anderson and Tweney (1997) demon- used as an analytictool to demonstrate the following con- strated a degree of power-curve emergence in non-range- sequences of arithmetic averaging: (1) For exponential limited distributions. component functions, the aggregate curve has an en- hanced fit to a power function (e.g., Point P); (2) for range- Conclusions limited linear and logarithmic components, power curves The present simulations and RSAs support the hypoth- can emerge if some but not all components contain a Y2 esis that component curves with highly variable slopes

Figure 3. Mean R2 as a function of intercomponent slope variability (the standard deviation of the random deviate distribution) and of fit type. Each panel shows simulation results for a particular type of component curve. POWER LAW 1067

Figure 4. Two-dimensional response surface analysis. Point P represents the arithmetic average of two exponential curves. Point Q represents the arithmetic average of two range-limited logarithmic curves. tend to produce approximationsto power curves at the ag- permutations in J. R. Anderson, 1982, 1993), cognitive gregate level—whether those componentslopes are expo- and behavioralsubprocedures,or productions,can operate nential,range-limitedlinear, or range-limitedlogarithmic. sequentially with one another and can thereby produce In addition,the simulationsshow that when the components additive effects on completion time for a given task. Con- themselves are power curves, the aggregate curve tends to sequently, insofar as different productions might possess be either power or logarithmic, depending on the amount very different learning rates, the power law of learning of variability and depending on the value of the A para- could arise solely from variable learning rates across pro- meter. ductions. Indeed, Newell and Rosenbloom (1981) sug- This conclusiondoes not deny the power function’sde- gested such production variability as one possible expla- scriptive utility.Instead,the function can be reconceptual- nation for the power law. The present findings bolster the ized as indicating a much deeper generalization: That is, plausibility of such an explanation, given that power- it may be that the power law is ubiquitous because slope curve emergence seems not to require that the component variability is ubiquitous.Moreover, occasionalor even fre- curves have a particularmathematicalform (e.g., the com- quent violations of the power law (i.e., exponential for- ponents do not have to be exponential).The evidencethus getting, Wickelgren, 1972; or exponential skill acquisi- far suggeststhat sufficient slope variability(and/or the re- tion, Heathcote, Brown, & Mewhort, 2000) need not sultantdistortionof the distributionof slopes), along with undermine the power function’s validity. Indeed, adher- appropriate range limitation of components, may be pri- ence to the power law might turn out to be valuable as mary preconditionsfor power-curve emergence. an index of the magnitude of intercomponent slope variability. REFERENCES Power-curve emergence also has clear theoretical sig- nificance, particularly if the componentcurves are aggre- Anderson, J. R. (1982). Acquisition of cognitive skill. Psychological gated within a system rather than across subjects or ex- Review, 89, 369-406. Anderson, J. R. (1993). Rules of the mind. Hillsdale, NJ: Erlbaum. perimental conditions. Anderson, J. R., & Schooler, L. J. (1991). Reflections of the envi- For example, a multiple-trace memory system, such as ronment in memory. Psychological Science, 2, 396-408. Minerva 2 (Hintzman, 1986,1988), will likelyexhibitdif- Anderson, R. B., & Tweney, R. D. (1997). Artifactual power curves in ferent decay rates for different traces. Moreover, the Mi- forgetting. Memory & Cognition, 25, 724-730. Anderson, R. B., Tweney, R. D., Rivardo, M., & Duncan, S. nerva 2 model additively combines individual traces into (1997). Need probabilityaffects retention: A direct demonstration.Memory & theoretically meaningful composites (Hintzman called Cognition, 25, 867-872. them “echoes”). Thus, a power law of forgetting might Estes, W. K. (1956). The problem of inference from curves based on arise from the additive combination of forgetting curves group data. Psychological Bulletin, 53, 134-140. across multiple traces. Heathcote, A., Brown, S., & Mewhort, D. J. K. (2000). The power law repealed: The case for an exponential law of practice. Psycho- Emergent power curves also have theoretical signifi- nomic Bulletin & Review, 7, 185-207. cance with regard to production systems in learning. For Hintzman, D. L. (1986). “Schema abstraction” in a multiple-trace example, in John Anderson’sACT theory (and its various memory model. Psychological Review, 93, 411-428. 1068 ANDERSON

Hintzman, D. L. (1988). Judgments of frequency and recognition mem- NOTES ory in a multiple-trace memory model. Psychological Review, 95, 528-551. 1. For purposes of simplicity and consistency, the term slope is used, Logan, G. D. (1988).Toward an instance theory of automatization. Psy- throughout, to refer to the B parameter. chological Review, 95, 492-527. 2.Here and throughout,the term form refers to a curve’s general math- Myung I. J., Kim, C., & Pitt, M. A. (2000). Toward an explanation of ematical form (e.g., exponential, logarithmic)—not the precise form that the power law artifact: Insights from response surface analysis. Mem- includes specific values for the parameters. ory & Cognition, 28, 832-840. 3. The arithmetic averaging of non-range-limited logarithmic curves Neves, D. M., & Anderson, J. R. (1981). Knowledge compilation: produces a logarithmic aggregate curve (Estes, 1956). In addition, geo- Mechanisms for the automatization of cognitive skills. In J. R. An- metric averaging of exponential-curve or power-curve components derson (Ed.), Cognitive skills and their acquisition (pp. 57-84). Hills- preserves their general mathematical form so that the aggregate curve dale, NJ: Erlbaum. is also an exponential curve (R. B. Anderson & Tweney, 1997; Estes, Newell, A., & Rosenbloom, P. S. (1981). Mechanisms of skill acqui- 1956). sition and the law of practice. In J. R. Anderson (Ed.), Cognitive skills 4. Though the present simulations focused exclusively on forgetting, and their acquisition (pp. 1–55). Hillsdale, NJ: Erlbaum. the results are applicable to any domain in which behavior varies con- Rubin, D. C. (1982). On the retention function for autobiographical tinuously as a function of some independent variable. Such domains in- memory. Journal of Verbal Learning & Verbal Behavior, 21, 21-38. clude other areas of psychology(e.g.,skill acquisitionand psychophysics) Rubin, D. C., & Wenzel, A.E. (1996).One hundred years of forgetting: and areas within economics. The findings may also have applicability in A quantitative description of retention. Psychological Review, 103, the biological and physical sciences. 734-760. 5. Having demonstrated the relevance of RSA to the present findings, Stevens, S. S. (1971). Neural events and the psychophysical law. Sci- I must point out that the simulations presented in the main body of this ence, 170, 1043-1050. paper involved fitting aggregate curves to two-parameter functions (the Wickelgren,W.A. (1972).Trace-resistance and the decay of long-term A parameter was fixed for each simulated component but was free to memory. Journal of Mathematical Psychology, 9, 418-455. vary in the fitting of the aggregate curves to mathematical functions). In Wickens, T.D. (1998). On the form of the retention function: Comment contrast, the two-dimensional RSA assumes one-parameter functions— on Rubin and Wenzel (1996): A qualitative description of retention. at both the component level and the aggregate level. Psychological Review, 105, 379-386. Wixted, J. T., & Ebbesen, E. B. (1991). On the form of forgetting. Psy- chological Science, 2, 409-415. Wixted, J. T., & Ebbesen, E. B. (1997). Genuine power curves in forgetting: A quantitative analysis of individual subject forgetting func- (Manuscript received May 12, 1999; tions. Memory & Cognition, 25, 731-739. revision accepted for publication April 28, 2001.)